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Introduction | ||
1 | Polynomial functions and combinatorics | 1 |
1.1 | Introductory remarks | 1 |
1.2 | Schur's thesis | 2 |
1.3 | The polynomial algebra | 9 |
1.4 | Combinatorics | 13 |
1.5 | Character theory and weight spaces | 17 |
1.6 | Irreducible objects in P[subscript K](n, r) | 22 |
2 | The Schur algebra | 25 |
2.1 | Definition | 26 |
2.2 | First properties | 28 |
2.3 | The Schur algebra S[subscript K](n,r) | 36 |
2.4 | Bideterminants and codeterminants | 38 |
2.5 | The Straightening Formula | 45 |
2.6 | The Desarmenien matrix and independence | 53 |
3 | Representation theory of the Schur algebra | 59 |
3.1 | Modules for [Alpha subscript r] and S[subscript r] | 60 |
3.2 | Schur modules as induced modules | 63 |
3.3 | Heredity chains | 67 |
3.4 | Schur modules and Weyl modules | 73 |
3.5 | Modular representation theory for Schur algebras | 84 |
4 | Schur functors and the symmetric group | 89 |
4.1 | The Schur functor | 90 |
4.2 | Applying the Schur functor | 94 |
4.3 | Hom functors for quasi-hereditary algebras | 107 |
4.4 | Decomposition numbers for G and [Gamma] | 111 |
4.5 | [Delta]-[actual symbol not reproducible]-good filtrations | 116 |
4.6 | Young modules | 122 |
5 | Block theory | 130 |
5.1 | Summary of block theory | 131 |
5.2 | Return of the Hom functors | 137 |
5.3 | Primitive blocks | 138 |
5.4 | General blocks | 143 |
5.5 | The finiteness theorem | 146 |
5.6 | Examples | 151 |
6 | The q-Schur algebra | 159 |
6.1 | Quantum matrix space | 161 |
6.2 | The q-Schur algebra, first visit | 164 |
6.3 | Weights and polynomial modules | 166 |
6.4 | Characters and irreducible [Alpha subscript q](n)-modules | 168 |
6.5 | R-forms for q-Schur algebras | 170 |
6.6 | The q-Schur algebra, second visit | 173 |
7 | Representation theory of S[subscript q](n,r) | 185 |
7.1 | q-Weyl modules | 186 |
7.2 | The q-determinant in [Alpha subscript q](n,r) | 192 |
7.3 | A quantum GL[subscript n] | 195 |
7.4 | The category P[subscript q](n,r) | 199 |
7.5 | P[subscript q](n,r) is a highest weight category | 200 |
7.6 | Representations of GL[subscript n](q) and the q-Young modules | 202 |
7.7 | Conclusion | 210 |
Appendix: a review of algebraic groups | 212 | |
A.1 Linear algebraic groups: definitions | 212 | |
A.2 Examples of linear algebraic groups | 213 | |
A.3 The weight lattice | 214 | |
A.4 Root systems | 214 | |
A.5 Weyl groups | 215 | |
A.6 The affine Weyl group | 216 | |
A.7 Simple modules for reductive groups | 217 | |
A.8 General linear group schemes | 218 | |
References | 220 | |
Index of Notation | 227 | |
Index of Terms | 229 |
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